Linear Equations

NZ Level 7 (NZC) Level 2 (NCEA)

Geometrical Problems with Coordinates

Lesson

We've looked at how to plot straight lines on the number plane. Now we are going to look at how to plot a series of coordinates to create geometric shapes.

Here are some helpful formulae and properties that will help us solve these kinds of problems:

- Distance formula: $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$
`d`=√(`x`2−`x`1)2+(`y`2−`y`1)2 - Gradient formula: $m=\frac{y_2-y_1}{x_2-x_1}$
`m`=`y`2−`y`1`x`2−`x`1 - Mid-point formula: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$(
`x`1+`x`22,`y`1+`y`22) - Parallel lines have equal gradient: $m_1=m_2$
`m`1=`m`2 - The product of the gradients of perpendicular lines is $-1$−1: $m_1m_2=-1$
`m`1`m`2=−1

Remember!

Different shapes have different properties.

These can be used to help plot and identify features of shapes on a number plane, so make sure you're familiar with the properties of different triangles and quadrilaterals.

$A$`A`$\left(-2,-1\right)$(−2,−1), $B$`B`$\left(0,0\right)$(0,0) and $C$`C`$\left(1,k\right)$(1,`k`) are the vertices of a right-angled triangle with right angle at $B$`B`.

Find the value of $k$

`k`.Find the area of the triangle.

Given Line P: $y=-6x-4$`y`=−6`x`−4, Line Q: $y=\frac{x}{6}+6$`y`=`x`6+6, Line R: $y=-6x-1$`y`=−6`x`−1 and Line S: $y=\frac{x}{6}+1$`y`=`x`6+1.

Complete the following:

$m$

`m`_{P }= $\editable{}$$m$

`m`_{Q }= $\editable{}$$m$

`m`_{P}x $m$`m`_{Q }= $\editable{}$Complete the following:

$m$

`m`_{Q }= $\editable{}$$m$

`m`_{R }= $\editable{}$$m$

`m`_{Q}x $m$`m`_{R }= $\editable{}$Complete the following:

$m$

`m`_{R }= $\editable{}$$m$

`m`_{S }= $\editable{}$$m$

`m`_{R}x $m$`m`_{S }= $\editable{}$Complete the following:

$m$

`m`_{S }= $\editable{}$$m$

`m`_{P }= $\editable{}$$m$

`m`_{S}x $m$`m`_{P }= $\editable{}$What type of quadrilateral is formed by lines: P, Q, R, and S?

Trapezoid

ARectangle

BRhombus

CParallelogram

DTrapezoid

ARectangle

BRhombus

CParallelogram

D

Apply co-ordinate geometry techniques to points and lines

Form and use linear, quadratic, and simple trigonometric equations

Apply co-ordinate geometry methods in solving problems

Apply algebraic methods in solving problems